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TR10-198 | 13th December 2010 19:55

Parameterized Bounded-Depth Frege is Not Optimal



A general framework for parameterized proof complexity was introduced by Dantchev, Martin, and Szeider (FOCS'07). In that framework the parameterized version of any proof system is not fpt-bounded for some technical reasons, but we remark that this question becomes much more interesting if we restrict ourselves to those parameterized contradictions $(F,k)$ in which $F$ itself is a contradiction. We call such parameterized contradictions strong, and with one important exception (vertex cover) all interesting contradictions we are aware of are strong. It follows from the gap complexity theorem of Dantchev et al. that tree-like Parameterized Resolution is not fpt-bounded w.r.t. strong parameterized contradictions.

The main result of this paper significantly improves upon this by showing that even the parameterized version of bounded-depth Frege is not fpt-bounded w.r.t. strong contradictions. More precisely, we prove that the pigeonhole principle requires proofs of size $n^{\Omega(k)}$ in bounded-depth Frege, and, as a special case, in dag-like Parameterized Resolution. This answers an open question posed by Dantchev et al.

In the opposite direction, we interpret a well-known FPT algorithm for vertex cover as a DPLL procedure for Parameterized Resolution. Its generalization leads to a proof search algorithm for Parameterized Resolution that in particular shows that tree-like Parameterized Resolution allows short refutations of all parameterized contradictions given as bounded-width CNF's.

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